OP-00494 Open

A periodic nonsingular billiard trajectory in every triangle

Note: Giovanni Forni claims a periodic orbit for every polygon in “Existence of a Periodic Orbit for Billiards in Polygons” (arXiv:2606.10102, 2026). The registry’s automated review and an independent check found that the proof does not establish it: its lower bound on the Betti number of the cut locus fails on explicit examples.

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Let $D \subset \mathbb{R}^2$ be a nondegenerate closed triangle. A periodic billiard trajectory in $D$ is given by an integer $m \ge 2$ and points $q_0, \dots, q_{m-1}$, indices taken modulo $m$, each in the relative interior of a side, with $q_i \ne q_{i+1}$, such that each segment $[q_i, q_{i+1}]$ lies in $D$ and at each $q_i$ the incoming and outgoing unit velocities are related by reflection across the side containing $q_i$. Vertices and grazing collisions are excluded, and both the position and the direction repeat.

Fagnano showed in 1775 that every acute triangle has one, its orthic triangle, and every right triangle has one as well. Masur proved that every rational polygon has a periodic billiard trajectory, and Boshernitzan, Galperin, Krüger and Troubetzkoy that they are dense. Schwartz proved that every triangle sufficiently close to the $30$-$60$-$90$ triangle has one (2006) and, with a computer-assisted proof, every triangle whose angles are all at most $100$ degrees (2009); Hooper and Schwartz proved it for every sufficiently small perturbation of an isosceles triangle. A 2018 computer-assisted preprint of Tokarsky, Garber, Marinov and Moore extends the $100$-degree result to obtuse angles of at most $112.3$ degrees.

Problem. Does every nondegenerate triangle $D \subset \mathbb{R}^2$ have a periodic billiard trajectory in this sense?

Source: A classical question, going back at least to Fagnano's periodic path in acute triangles (1775); stated as the Triangular Billiards Conjecture by Schwartz (2006; 2009, Conjecture 1.1).

Imported from github.com/MColbrook/AIM.

Classification

Primary: DS — Dynamical Systems
Keywords: AIM Open Applied Problems, periodic orbits, polygonal billiards, triangles

References

@article{schwartz2006,
  author  = {Schwartz, Richard Evan},
  title   = {Obtuse triangular billiards {I}: near the $(2,3,6)$ triangle},
  journal = {Experiment. Math.},
  volume  = {15},
  number  = {2},
  year    = {2006},
  pages   = {161--182},
  doi     = {10.1080/10586458.2006.10128961},
}
@article{schwartz2009,
  author  = {Schwartz, Richard Evan},
  title   = {Obtuse triangular billiards {II}: one hundred degrees worth of periodic trajectories},
  journal = {Experiment. Math.},
  volume  = {18},
  number  = {2},
  year    = {2009},
  pages   = {137--171},
  doi     = {10.1080/10586458.2009.10128891},
}
@article{masur1986,
  author  = {Masur, Howard},
  title   = {Closed trajectories for quadratic differentials with an application to billiards},
  journal = {Duke Math. J.},
  volume  = {53},
  number  = {2},
  year    = {1986},
  pages   = {307--314},
  doi     = {10.1215/S0012-7094-86-05319-6},
}
@article{boshernitzan1998,
  author  = {Boshernitzan, M. and Galperin, G. and Kr{\"u}ger, T. and Troubetzkoy, S.},
  title   = {Periodic billiard orbits are dense in rational polygons},
  journal = {Trans. Amer. Math. Soc.},
  volume  = {350},
  number  = {9},
  year    = {1998},
  pages   = {3523--3535},
  doi     = {10.1090/S0002-9947-98-02089-3},
}
@article{hooper2009,
  author  = {Hooper, W. Patrick and Schwartz, Richard Evan},
  title   = {Billiards in nearly isosceles triangles},
  journal = {J. Mod. Dyn.},
  volume  = {3},
  number  = {2},
  year    = {2009},
  pages   = {159--231},
  doi     = {10.3934/jmd.2009.3.159},
}
@misc{tokarsky2018,
  author        = {Tokarsky, George and Garber, Jacob and Marinov, Boyan and Moore, Kenneth},
  title         = {One hundred and twelve point three degree theorem},
  year          = {2018},
  eprint        = {1808.06667},
  archivePrefix = {arXiv},
}
@misc{aim-periodic-triangular-billiard,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{A periodic nonsingular billiard trajectory in every triangle}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/250-periodic-triangular-billiard.md},
  note         = {Entry 250 at commit a153f46, 2026-10-13},
}

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Open	—	Listed open with a note: the claimed proof was found wrong.

Citing this problem

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OP-00494, "A periodic nonsingular billiard trajectory in every triangle", clopen.solutions.